I have two coupled resonators with complex resonance frequencies of $\beta_1$ and $\beta_2$. The coupling between them is varying sinusoidal with time. The time-evolution of their resonant mode field ($A,B$) can be written as:
$$\frac{dA}{dt} = \iota\beta_1 A + \iota [\gamma\sin(\Omega t)]) B$$
$$\frac{dB}{dt} = \iota\beta_2 B + \iota [\gamma\sin(\Omega t) ] A$$
I want to get a comprehensive solution of this differential equation. I tried to look for books where periodic differential equations are explained but no success so far. Any idea how to find an exact solution of the differential equations above? $\beta_1$, $\beta_2$, $\Omega$, and $\gamma$s are constants.
Write your system under its matrix form :
$$\pmatrix{\frac{dA}{dt} \\ \frac{dB}{dt} }= \iota \underbrace{\pmatrix{\beta_1 & \gamma\sin(\Omega t)\\\gamma\sin(\Omega t) & \beta_2}}_M\pmatrix{A\\B}$$
whose solution (extension of the classical scalar ODE : $y'(t)=my(t) \implies y(t)=\exp(mt).y(0) $) is
$$\pmatrix{A\\B}=\exp(\iota M)\pmatrix{A_0\\B_0}$$
where $A(0),B(0)$ are the initial values.
If you work with Matlab, matrix exponentiation is done by "expm" function.